Fixed Income Quantitative Credit Research March 2004 Credit Spreads Explained Dominic O'Kane and Saurav Sen Credit investors need a measure to determine how much they are being paid to compensate them for assuming the credit risk embedded within a security. A number of such measures exist, and are commonly known as credit spreads since they attempt to measure the return of the credit asset relative to some higher credit quality benchmark. Each has its own strengths and weaknesses. In this article, we define, describe and analyse the main credit spreads for fixed rate bonds , floating rate notes and the credit default swap. Lehman Brothers | Quantitative Credit Research Quarterly Credit Spreads Explained Dominic O’Kane Credit investors need a measure to determine how much they are being paid to compensate +44 20-7102-2628 them for assuming the credit risk embedded within a security. A number of such measures [email protected] exist, and are commonly known as credit spreads since they attempt to measure the return of the credit asset relative to some higher credit quality benchmark. Each has its own strengths Saurav Sen and weaknesses. In this article, we define, describe and analyse the main credit spreads for 1 +44 20-7102-2940 fixed rate bonds , floating rate notes and the credit default swap. [email protected] 1. INTRODUCTION It is useful for credit investors to have a measure to determine how much they are being paid to compensate them for assuming the credit risk embedded within a security. This credit risk may be embedded within a bond issued by some corporate or sovereign or may be synthesized through some credit derivative. Such a measure of credit quality should enable comparison between securities issued by a company, which may differ in terms of maturity, coupon or seniority. It should also facilitate comparisons with the securities of other issuers. Ultimately it should enable the credit investor to identify relative value opportunities within the bonds of a single issuer, and across different issuers. When we enlarge our universe of credit instruments to include not just fixed rate bonds, but also floating rate bonds, asset swaps and default swaps, it is only natural to try to define a credit risk metric which also allows a comparison across instruments. For example, we would like to know when a credit default swap is priced fairly relative to a cash bond when both are linked to the same issuer. This is especially important for determining the relative value of a default swap basis trade. While we would like one simple credit measure, there is in fact a multiplicity of such measures. Most are called “credit spreads” since they attempt to capture the difference in credit quality by measuring the return of the credit risk security as a spread to some higher credit quality benchmark, typically either the government (assumed credit risk free) curve or the same maturity Libor swap rate (linked to the funding rate of the AA-rated commercial banking sector). Well-known credit measures include the yield spread, the asset swap spread, the option adjusted spread (OAS), the zero volatility spread, the discount margin, the default swap spread and the hazard rate. Each is defined in its own particular way and so has its own corresponding strengths and weaknesses. However, since they play such a fundamental role in the trading, analysis and valuation of credit securities, it is essential that there exist a clear picture as to the information contained in each of these different credit spreads. The purpose of this report is to define, explain and examine these different credit spreads. We would also like to understand the relationship between different credit spread measures. To do this we have to set up a unifying credit risk modelling framework through which we can express each of these credit-spread measures. Such a model will enable us to understand how different measures behave with changing credit quality and asset indicatives. A comparison of the various credit spread measures is left to a forthcoming Quantitative Credit Research Quarterly article. 1 An analysis of spread measures within the context of agency bonds has been published in Tuckman (2003). 15 March 2004 Please see important analyst certification(s) on the back cover of this report 1 Lehman Brothers | Quantitative Credit Research Quarterly The remainder of this paper is organised as follows: We describe a number of credit spread measures in turn, starting with those for fixed coupon cash bonds, followed by those for floating rate bonds. We focus mainly on the spread measures quoted by Bloomberg on its YAS, YAF and ASW pages and also those used by Lehman Brothers on Lehman Live. Additionally, we define and explain the credit default swap spread. 2. CREDIT SPREAD MEASURES FOR FIXED RATE BONDS We start by discussing the most common credit spread measures for fixed rate bonds. THE YIELD SPREAD The yield spread, also known as the yield-yield spread, is probably the most widely used credit spread measure used by traders of corporate bonds. Its advantage is that it is simply the difference between two yields – that of the credit bond and that of the associated treasury benchmark and so is easy to compute and sufficiently transparent that it is often used as the basis to price in the closing of a crossing trade of credit bond versus treasury bond. Definition The yield spread is the difference between the yield-to-maturity of the credit risky bond and the yield-to-maturity of an on-the-run treasury benchmark bond with similar but not necessarily identical maturity. The mathematical definition of the yield-to-maturity is well known and has been discussed at length elsewhere (Fabozzi 2003). However we repeat it here for the sake of completeness. It is the constant discounting rate which, when applied to the bond’s cashflows, reprices the bond. If we denote the full (including accrued) price of the defaultable bond by Pfull, the annualised coupon by CD, the coupon frequency by fD, and the time to each of the cash flow payments in years by T1, …, TN, then the yield yD of the defaultable bond is the solution to the following equation. C / f C / f 100 + C / f P full = D D + D D +...+ D D + f DT1 + f DT2 + f DTN (1 yD / f D ) (1 yD / fD ) (1 yD / f D ) Note that the full coupon is used for the first period, consistent with using the full price. The times to the payments are calculated using the appropriate day count convention, such as 30/360 (bond) or ACT. A 1-dimensional root-searching algorithm is typically used to find the value of yD which satisfies this equation. Before we can consider the information content of the yield spread, let us consider for a moment the assumptions behind the yield to maturity measure. These are: 1. An investor who buys this asset can only achieve a return equal to the yield measure if the bond is held to maturity and if all coupons can be reinvested at the same rate as the yield. In practice, this is not possible since changes in the credit quality of the issuer may cause yields to change through time. As many investors may re-invest coupons at rates closer to LIBOR, at least temporarily, the realised return will usually be lower than the yield to maturity. 2. It assumes that the yield curve is flat which is not generally true. In practice, we would expect different reinvestment rates for different maturities. In the yield to maturity, these reinvestment rates are the same for all maturities. 15 March 2004 2 Lehman Brothers | Quantitative Credit Research Quarterly To calculate the yield spread we also need to calculate the yield of the benchmark government bond yB as above. The yield spread is then given by the following relationship: = − Yield Spread yD yB . Example As an example, consider the following bond: Ford Motor Credit 7.25% 25 Oct 2011 which priced at 107.964 on the 9th February 2004, and settles on the 12th February 2004. This bond has semi-annual coupons which accrue on 30/360 (Bond) basis. With 107 days2 of accrued interest worth 2.1549, the full price of the bond is 110.1189. A simple price-yield calculation, summarised in Table 1 below, gives a yield of 5.94%. The benchmark for this corporate bond at issuance was the 10-yr on-the-run treasury, with 5% coupon and maturity date 15 August 2011. As the bond has rolled down the curve, the current benchmark is the 5-yr on-the-run treasury, which has 3% coupon and matures on 15 February 2009. It has a yield to maturity of 3.037%. As the yield of the Ford bond is 5.94% and that of the benchmark is 3.04%, so the yield spread is 290bp. In this case, therefore, there is a maturity mismatch between the bond and its benchmark where the Ford bond matures almost 3 years after the benchmark. The benchmark has also changed since the bond was issued. Table 1. Yield to maturity calculation summary Cashflow on Time in Years Yield Discount Cashflow Date $100 Notional Yield (30/360 Basis) Factor PV 12-Feb-04 0.008 0.9995 25-Apr-04 3.625 5.94% 0.211 0.9877 3.5805 25-Oct-04 3.625 5.94% 0.711 0.9593 3.4773 25-Apr-05 3.625 5.94% 1.211 0.9316 3.3771 25-Oct-05 3.625 5.94% 1.711 0.9048 3.2797 25-Apr-06 3.625 5.94% 2.211 0.8787 3.1852 25-Oct-06 3.625 5.94% 2.711 0.8533 3.0934 25-Apr-07 3.625 5.94% 3.211 0.8288 3.0042 25-Oct-07 3.625 5.94% 3.711 0.8049 2.9176 25-Apr-08 3.625 5.94% 4.211 0.7817 2.8335 25-Oct-08 3.625 5.94% 4.711 0.7591 2.7519 25-Apr-09 3.625 5.94% 5.211 0.7373 2.6725 25-Oct-09 3.625 5.94% 5.711 0.7160 2.5955 25-Apr-10 3.625 5.94% 6.211 0.6954 2.5207 25-Oct-10 3.625 5.94% 6.711 0.6753 2.4480 25-Apr-11 3.625 5.94% 7.211 0.6559 2.3775 25-Oct-11 103.625 5.94% 7.711 0.6370 66.0041 Yield-Implied Full Price of the Bond: 110.1189 Interpretation We can make a number of observations about the yield spread as a credit risk measure: • It shares all of the weaknesses of the yield to maturity measure in terms of constant reinvestment rate and hold to maturity.